Computational Methods Based on the Guaranteed Capture Basin Algorithm
This chapter is devoted to numerical approximation methods for managing replicating portfolios and more complex financial instruments. One aim is to regulate evolutions under uncertainty in order not only to reach a target in finite time but also to fulfi
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Computational Methods Based on the Guaranteed Capture Basin Algorithm
17.1 Guaranteed Capture Basin Method for Evaluating Portfolios 17.1.1 Classical Option Evaluation The fundamental problem arising in the framework of dynamic replicating portfolios is to determine a hedging strategy p(·) such that, whatever the uncertain evolution of the underlying asset price S(·) is, a payoff is realized at the exercise time or at any time before, depending on the type of option. This can be formalized in terms of viability and target capturability in the presence of uncertainty, guaranteeing the capture of a target. This is the main issue of viability theory. Therefore, one can deduce from the mathematical and geometrical properties of capture basins the optimal rules for managing complex financial instruments. Furthermore, once discretized in this natural formulation, the Guaranteed Capture Basin Algorithm provides the valuation of an optimal portfolio and its management under different representations of uncertainty. We refer the reader to Chap. 18 by Jean-Pierre Aubin, Luxi Chen, and Olivier Dordan. A put or call is an agreement conferring the right to sell (put) or buy (call) a quantity of an asset at a given date (European put or call) or at any date before a fixed date T (American put or call). We aim at determining the value of the agreement at the start. This value is the price the seller should ask for to protect herself against risk. It measures the cost of risk covering. Facing the risks inherent in her position, the seller builds up a theoretical portfolio by investing in the underlying asset through self-financing. This permanently adjusted portfolio yields the same losses and profits as the put or call. It is said to replicate the put or call. For European “vanilla” options we know that the Guaranteed Capture Basin Algorithm provides both approximated values of the option depending on the refinement level of the discretization, on the maturity and price of the underlying asset, and on the option hedging strategy. In cases where uncertainty is characterized P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 17, © Springer Science+Business Media New York 2013
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17 Computational Methods Based on the Guaranteed Capture Basin Algorithm
by a geometric Brownian motion leading to the Black–Scholes option pricing formula and considering approximation of this Brownian motion by random walk following the Cox, Ross, and Rubinstein (CRR) discrete approach, the Guaranteed Capture Basin Algorithm returns classic valuation results but also provides valuation and hedging strategies when considering other types of uncertainty such as tychastic uncertainty.
17.1.2 Limits of Classic Evaluation Methods Taking into account transaction costs in the framework of the Black–Scholes approach becomes intricate [23,126]. The Guaranteed Capture Basin Method can be applied to handle any constraints on asset prices or share q
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